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P. Tilli

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Open access Jul 2026

The Isoperimetric Inequality for Partial Sums of Toeplitz Eigenvalues in the Fock Space

Abstract. We prove that, among all subsets [Formula: see text] having circular symmetry and prescribed measure, the ball is the only maximizer of the sum of the first [Formula: see text] eigenvalues ([Formula: see text]) of the corresponding Toeplitz operator [Formula: see text] on the Fock space [Formula: see text]. As a byproduct, we prove that, again among circularly symmetric sets of prescribed measure, balls maximize any Schatten [Formula: see text]-norm of [Formula: see text] for [Formula: see text] (and minimize the corresponding quasinorm for [Formula: see text]), and that the second eigenvalue is maximized by a particular annulus. Moreover, we extend some of these results to general radial symbols in [Formula: see text] with [Formula: see text], characterizing those that maximize the sum of the first [Formula: see text] eigenvalues. We also show a symmetry breaking phenomenon for the second eigenvalue, when the assumption of circular symmetry is dropped.

Fabio Nicola, Federico Riccardi, P. Tilli · SIAM Journal on Mathematical... · 0 citations

Preprint Jul 2026

On the existence of extremizers for the sum of eigenvalues of Toeplitz operators

We prove that, among all measurable sets $\Omega\subset\mathbb{C}$ of prescribed Lebesgue measure, there exists a set maximizing the sum of the first $K$ eigenvalues ($K\geq 1$) of the associated Toeplitz operator on the Fock space. In the Fock setting, the case $K=1$ is well known, the optimal sets being balls of prescribed measure, whereas for $K>1$ the existence of optimal sets appears to be new (maximizers are not known explicitly, and the optimality of balls remains conjectural). Moreover, under mild assumptions, our proof extends to localization operators associated with abstract wavelet transforms. In this broader setting, the result is new even for $K=1$. As an application, we prove the existence of optimal sets for the Donoho--Stark concentration problem and its generalization to orthonormal systems.

Fabio Nicola, Federico Riccardi, P. Tilli · 2 citations · ⚡1